Revision as of 14:27, 7 December 2009

Implication is a logical operation on two statements, typically represented by the variables P and Q. "P implies Q" is written symbolically as "P → Q". Equivalent statements include "if P then Q", "P is sufficient for Q", and "Q is necessary for P".

Definition

Note that if P is true, then Q must also be true for the implication to hold (be true). However, if P is false, Q may or may not be true and the implication still holds.

To illustrate the latter fact, consider a teacher who tells her class that any student who gets 100% on the final exam will pass the class. In other words:

P: A student gets 100% on the final.

Q: That student passes the class.

P → Q: If a student gets 100% on the final, then that student passes the class.

Now consider the case in which two students do poorly on the final exam; one of them did well enough on the other exams to pass the course, but the other did not. Did the teacher lie?

No. She said nothing about students who do not get 100% on the final (i.e., the case where P is false). Unless there is a student who both got 100% on the final and did not pass the course, the teacher told the truth.

For this reason, "P → Q" can be restated as "¬(P ∧ ¬ Q)" or "not (P and not Q)", or "it is not the case that P is true and Q is false".